# Properties

 Label 2475.2.f.d Level $2475$ Weight $2$ Character orbit 2475.f Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 495) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + \beta q^{7} -3 q^{8} +O(q^{10})$$ $$q + q^{2} - q^{4} + \beta q^{7} -3 q^{8} + ( 3 + \beta ) q^{11} -2 \beta q^{13} + \beta q^{14} - q^{16} + 2 q^{17} + 3 \beta q^{19} + ( 3 + \beta ) q^{22} -2 \beta q^{26} -\beta q^{28} -6 q^{29} + 5 q^{32} + 2 q^{34} -6 q^{37} + 3 \beta q^{38} -6 q^{41} + \beta q^{43} + ( -3 - \beta ) q^{44} + 6 \beta q^{47} + 5 q^{49} + 2 \beta q^{52} + 3 \beta q^{53} -3 \beta q^{56} -6 q^{58} + 6 \beta q^{59} + 6 \beta q^{61} + 7 q^{64} -2 q^{68} + 6 \beta q^{71} + 8 \beta q^{73} -6 q^{74} -3 \beta q^{76} + ( -2 + 3 \beta ) q^{77} + 9 \beta q^{79} -6 q^{82} + 14 q^{83} + \beta q^{86} + ( -9 - 3 \beta ) q^{88} -5 \beta q^{89} + 4 q^{91} + 6 \beta q^{94} -12 q^{97} + 5 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{4} - 6q^{8} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{4} - 6q^{8} + 6q^{11} - 2q^{16} + 4q^{17} + 6q^{22} - 12q^{29} + 10q^{32} + 4q^{34} - 12q^{37} - 12q^{41} - 6q^{44} + 10q^{49} - 12q^{58} + 14q^{64} - 4q^{68} - 12q^{74} - 4q^{77} - 12q^{82} + 28q^{83} - 18q^{88} + 8q^{91} - 24q^{97} + 10q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2276.1
 − 1.41421i 1.41421i
1.00000 0 −1.00000 0 0 1.41421i −3.00000 0 0
2276.2 1.00000 0 −1.00000 0 0 1.41421i −3.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2475.2.f.d 2
3.b odd 2 1 2475.2.f.a 2
5.b even 2 1 2475.2.f.b 2
5.c odd 4 2 495.2.d.b yes 4
11.b odd 2 1 2475.2.f.a 2
15.d odd 2 1 2475.2.f.c 2
15.e even 4 2 495.2.d.a 4
33.d even 2 1 inner 2475.2.f.d 2
55.d odd 2 1 2475.2.f.c 2
55.e even 4 2 495.2.d.a 4
165.d even 2 1 2475.2.f.b 2
165.l odd 4 2 495.2.d.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.d.a 4 15.e even 4 2
495.2.d.a 4 55.e even 4 2
495.2.d.b yes 4 5.c odd 4 2
495.2.d.b yes 4 165.l odd 4 2
2475.2.f.a 2 3.b odd 2 1
2475.2.f.a 2 11.b odd 2 1
2475.2.f.b 2 5.b even 2 1
2475.2.f.b 2 165.d even 2 1
2475.2.f.c 2 15.d odd 2 1
2475.2.f.c 2 55.d odd 2 1
2475.2.f.d 2 1.a even 1 1 trivial
2475.2.f.d 2 33.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2} - 1$$ $$T_{29} + 6$$ $$T_{37} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$2 + T^{2}$$
$11$ $$11 - 6 T + T^{2}$$
$13$ $$8 + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$18 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$2 + T^{2}$$
$47$ $$72 + T^{2}$$
$53$ $$18 + T^{2}$$
$59$ $$72 + T^{2}$$
$61$ $$72 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$72 + T^{2}$$
$73$ $$128 + T^{2}$$
$79$ $$162 + T^{2}$$
$83$ $$( -14 + T )^{2}$$
$89$ $$50 + T^{2}$$
$97$ $$( 12 + T )^{2}$$